We call the inequalities coming from Lemma REF  subsheaf inequalities. The reason for this is the following. Suppose that \(\mathcal {O}_Z\)  is a \(G\) -cluster parameterised by a point in an irreducible exceptional divisor \(D\subset G\) -Hilb. Let \(\psi \)  be a character marking \(D\) . It follows from [1]} that \(r_\psi \)  is in the socle of \(H^0(\mathcal {O}_Z)\)  and is constant on \(D\) , hence defines a subsheaf \(S=\mathcal {O}_0\otimes \psi \)  of \(\mathcal {O}_Z\)  for all \(Z\in D\) . By definition we must have \(\theta (S)=\theta (\psi )>0\)  for all \(\theta \in \mathfrak {C}_0\)  and [1]} shows that this is indeed a wall of \(\mathfrak {C}_0\) .
